Hide wireframe in edit mode
Sadly this is not possible, however you can approximate a result like this fairly easily.

First, go to File > User Preferences > Themes > 3D View and change the Wire color to the default grey. (Around Hex 6B6B6B)



This will make it less visible, but not invisible. You can make it so that selected edges are not shown as well by going to the N menu and finding the Mesh Display options.



Also see this question